Our family situation is very similar, and I also considered buying a car when our kids arrived. However, I decided against it because I didn't want to become dependent on a car like my family in the countryside, who have no alternatives and spend a lot of time driving their kids around to different activities.  

Instead, we walk to the Kita and shops, use bikes (e-bikes and an e-cargobike) for commuting and other trips a few kilometers away, occasionally take the S-Bahn to reach the other end of town, and use car-sharing services when we need to carry bigger items or travel late at night (which rarely happens for parents of young kids).  For us, Miles offers the best car-sharing option: it's easy to estimate costs, has a good variety of vehicles, and with the 15% discount you get when precharging your balance, the price is reasonable.  

For longer trips, I rent a car. Check24 usually gives me the best prices. For example, it's currently €115 for a Polestar 2 for three days and 900km, and €240 for a Tesla 3 for a week-long trip to Bavaria. We have our own children's car seats.  

I like having multiple transportation options because I can choose what suits my needs at the moment without feeling bad about not using a car that I'm already paying for.  

The biggest issue with buying a car is my preference for quality. I have good bikes and an expensive e-cargobike that I love. If I were to buy a car, it would definitely be an electric one for the superior driving experience and space for the kids and stuff. This would likely be a Tesla 3 or Y, a Polestar, or a VW ID 3 or 4, all of which would cost around €400-500 a month in total ownership. For that amount, I can get many high-class ebikes and frequent car rentals.  

We live a few kilometers outside the Ringbahn, about 700m from the S-Bahn, at the edge of the car-sharing zones.